Identification of nonlinear systems using random impulse train inputs

Nonlinear systems that require discrete inputs can be characterized by using random impulse train (Poisson process) inputs. The method is analagous to the Wiener method for continuous input systems, where Gaussian white-noise is the input. In place of the Wiener functional expansion for the output of a continuous input system, a new series for discrete input systems is created by making certain restrictions on the integrals in a Volterra series. The kernels in the new series differ from the Wiener kernels, but also serve to identify a system and are simpler to compute. For systems whose impulse responses vary in amplitude but maintain a similar shape, one argument may be held fixed in each kernel. This simplifies the identification problem. As a test of the theory presented, the output of a hypothetical second order nonlinear system in response to a random impulse train stimulus was computer simulated. Kernels calculated from the simulated data agreed with theoretical predictions. The Poisson impulse train method is applicable to any system whose input can be delivered in discrete pulses. It is particularly suited to neuronal synaptic systems when the pattern of input nerve impulses can be made random.     

A nonlinear cascade model for action potential encoding in an insect sensory neuron.

Action potential encoding in the cockroach tactile spine neuron can be represented as a single-input single-output nonlinear dynamic process. We have used a new functional expansion method to characterize the nonlinear behavior of the neural encoder. This method, which yields similar kernels to the Wiener method, is more accurate than the latter and is efficient enough to obtain reasonable kernels in less than 15 min using a personal computer. The input stimulus was band-limited white Gaussian noise and the output consisted of the resulting train of action potentials, which were unitized to give binary values. The kernels and the system input-output signals were used to identify a model for encoding comprising a cascade of dynamic linear, static nonlinear, and dynamic linear components. The two dynamic linear components had repeatable and distinctive forms with the first being low-pass and the second being high-pass. The static nonlinearity was fitted with a fifth-order polynomial function over several input amplitude ranges and had the form of a half-wave rectifier. The complete model gave a good approximation to the output of the neuron when both were subjected to the same novel white noise input signal.     

Computational simulation of the input-output relationship in hippocampal pyramidal cells

The precise mapping of how complex patterns of synaptic inputs are integrated into specific patterns of spiking output is an essential step in the characterization of the cellular basis of network dynamics and function. Relative to other principal neurons of the hippocampus, the electrophysiology of CA1 pyramidal cells has been extensively investigated. Yet, the precise input-output relationship is to date unknown even for this neuronal class. CA1 pyramidal neurons receive laminated excitatory inputs from three distinct pathways: recurrent CA1 collaterals on basal dendrites, CA3 Schaffer collaterals, mostly on oblique and proximal apical dendrites, and entorhinal perforant pathway on distal apical dendrites. We implemented detailed computer simulations of pyramidal cell electrophysiology based on three-dimensional anatomical reconstructions and compartmental models of available biophysical properties from the experimental literature. To investigate the effect of synaptic input on axosomatic firing, we stochastically distributed a realistic number of excitatory synapses in each of the three dendritic layers. We then recorded the spiking response to different stimulation patterns. For all dendritic layers, synchronous stimuli resulted in trains of spiking output and a linear relationship between input and output firing frequencies. In contrast, asynchronous stimuli evoked non-bursting spike patterns and the corresponding firing frequency input-output function was logarithmic. The regular/irregular nature of the input synaptic intervals was only reflected in the regularity of output inter-burst intervals in response to synchronous stimulation, and never affected firing frequency. Synaptic stimulations in the basal and proximal apical trees across individual neuronal morphologies yielded remarkably similar input-output relationships. Results were also robust with respect to the detailed distributions of dendritic and synaptic conductances within a plausible range constrained by experimental evidence. In contrast, the input-output relationship in response to distal apical stimuli showed dramatic differences from the other dendritic locations as well as among neurons, and was more sensible to the exact channel densities. Action Editor: Alain Destexhe     

Some implications of the short-term synaptic plasticity for neuronal computation: a model study

Spikes arriving at the synaptic connection produce short-term plastic changes of the synaptic efficacy. Model experiments have shown that paired-pulse facilitation attaining its maximum after a specific interval between a pair of arriving spikes might turn a “weak” plastic synapse attached to an integrate-and-fire neuron to a frequency-tuned device. Resulting computational capabilities create biologically plausible mechanisms of information processing relating to: (i) real-time identification of temporal patterns in a stream of random spiking activity (a recognition problem); and (ii) codetermination of the specific activity routing among neurons (an addressing problem) resulting in definite spatio-temporal patterns of the output activity (an input-output pattern problem).     

Brain signal predictions from multi-scale networks using a linearized framework

Simulations of neural activity at different levels of detail are ubiquitous in modern neurosciences, aiding the interpretation of experimental data and underlying neural mechanisms at the level of cells and circuits. Extracellular measurements of brain signals reflecting transmembrane currents throughout the neural tissue remain commonplace. The lower frequencies (≲ 300Hz) of measured signals generally stem from synaptic activity driven by recurrent interactions among neural populations and computational models should also incorporate accurate predictions of such signals. Due to limited computational resources, large-scale neuronal network models (≳ 10⁶ neurons or so) often require reducing the level of biophysical detail and account mainly for times of action potentials (‘spikes’) or spike rates. Corresponding extracellular signal predictions have thus poorly accounted for their biophysical origin.Here we propose a computational framework for predicting spatiotemporal filter kernels for such extracellular signals stemming from synaptic activity, accounting for the biophysics of neurons, populations, and recurrent connections. Signals are obtained by convolving population spike rates by appropriate kernels for each connection pathway and summing the contributions. Our main results are that kernels derived via linearized synapse and membrane dynamics, distributions of cells, conduction delay, and volume conductor model allow for accurately capturing the spatiotemporal dynamics of ground truth extracellular signals from conductance-based multicompartment neuron networks. One particular observation is that changes in the effective membrane time constants caused by persistent synapse activation must be accounted for.The work also constitutes a major advance in computational efficiency of accurate, biophysics-based signal predictions from large-scale spike and rate-based neuron network models drastically reducing signal prediction times compared to biophysically detailed network models. This work also provides insight into how experimentally recorded low-frequency extracellular signals of neuronal activity may be approximately linearly dependent on spiking activity. A new software tool LFPykernels serves as a reference implementation of the framework.     

Short term synaptic depression imposes a frequency dependent filter on synaptic information transfer

Depletion of synaptic neurotransmitter vesicles induces a form of short term depression in synapses throughout the nervous system. This plasticity affects how synapses filter presynaptic spike trains. The filtering properties of short term depression are often studied using a deterministic synapse model that predicts the mean synaptic response to a presynaptic spike train, but ignores variability introduced by the probabilistic nature of vesicle release and stochasticity in synaptic recovery time. We show that this additional variability has important consequences for the synaptic filtering of presynaptic information. In particular, a synapse model with stochastic vesicle dynamics suppresses information encoded at lower frequencies more than information encoded at higher frequencies, while a model that ignores this stochasticity transfers information encoded at any frequency equally well. This distinction between the two models persists even when large numbers of synaptic contacts are considered. Our study provides strong evidence that the stochastic nature neurotransmitter vesicle dynamics must be considered when analyzing the information flow across a synapse.     

Predicting the synaptic information efficacy in cortical layer 5 pyramidal neurons using a minimal integrate-and-fire model

Synaptic information efficacy (SIE) is a statistical measure to quantify the efficacy of a synapse. It measures how much information is gained, on the average, about the output spike train of a postsynaptic neuron if the input spike train is known. It is a particularly appropriate measure for assessing the input–output relationship of neurons receiving dynamic stimuli. Here, we compare the SIE of simulated synaptic inputs measured experimentally in layer 5 cortical pyramidal neurons in vitro with the SIE computed from a minimal model constructed to fit the recorded data. We show that even with a simple model that is far from perfect in predicting the precise timing of the output spikes of the real neuron, the SIE can still be accurately predicted. This arises from the ability of the model to predict output spikes influenced by the input more accurately than those driven by the background current. This indicates that in this context, some spikes may be more important than others. Lastly we demonstrate another aspect where using mutual information could be beneficial in evaluating the quality of a model, by measuring the mutual information between the model’s output and the neuron’s output. The SIE, thus, could be a useful tool for assessing the quality of models of single neurons in preserving input–output relationship, a property that becomes crucial when we start connecting these reduced models to construct complex realistic neuronal networks.     

Correlation-distortion based identification of Linear-Nonlinear-Poisson models

Linear-Nonlinear-Poisson (LNP) models are a popular and powerful tool for describing encoding (stimulus-response) transformations by single sensory as well as motor neurons. Recently, there has been rising interest in the second- and higher-order correlation structure of neural spike trains, and how it may be related to specific encoding relationships. The distortion of signal correlations as they are transformed through particular LNP models is predictable and in some cases analytically tractable and invertible. Here, we propose that LNP encoding models can potentially be identified strictly from the correlation transformations they induce, and develop a computational method for identifying minimum-phase single-neuron temporal kernels under white and colored random Gaussian excitation. Unlike reverse-correlation or maximum-likelihood, correlation-distortion based identification does not require the simultaneous observation of stimulus-response pairs—only their respective second order statistics. Although in principle filter kernels are not necessarily minimum-phase, and only their spectral amplitude can be uniquely determined from output correlations, we show that in practice this method provides excellent estimates of kernels from a range of parametric models of neural systems. We conclude by discussing how this approach could potentially enable neural models to be estimated from a much wider variety of experimental conditions and systems, and its limitations.     

Nanoscale organization of the pre-synapse: Tracking the neurotransmitter release machinery

Chemical communication is underpinned by the fusion of neurotransmitter-containing synaptic vesicles with the plasma membrane at active zones. With the advent of super-resolution microscopy, the door is now opened to unravel the dynamic remodeling of synapses underpinning learning and memory. Imaging proteins with conventional light microscopy cannot provide submicron information vital to determining the nanoscale organization of the synapse. We will first review the current super-resolution microscopy techniques available to investigate the localization and movement of synaptic proteins and how they have been applied to visualize the synapse. We discuss the new techniques and analytical approaches have provided comprehensive insights into synaptic organization in various model systems. Finally, this review provides a brief update on how these super-resolution techniques and analyses have opened the way to a much greater understanding of the synapse, the fusion and compensatory endocytosis machinery.     

Effect of synaptic plasticity on sensory coding and steady-state filtering properties in the electric sense

Our modeling study examines short-term plasticity at the synapse between afferents from electroreceptors and pyramidal cells in the electrosensory lateral lobe (ELL) of the weakly electric fish Apteronotus leptorhynchus. It focusses on steady-state filtering and coherence-based coding properties. While developed for electroreception, our study exposes general functional features for different mixtures of depression and facilitation. Our computational model, constrained by the available in vivo and in vitro data, consists of a synapse onto a deterministic leaky integrate-and-fire (LIF) neuron. The synapse is either depressing (D), facilitating (F) or both (FD), and is driven by a sinusoidally or randomly modulated Poisson process. Due to nonlinearity, numerically computed input-output transfer functions are used to determine the filtering properties. The gain of the response at each sinusoidally modulated frequency is computed by dividing the fitted amplitudes of the input and output cycle histograms of the LIF models. While filtering is always low-pass for F alone, D alone exhibits a gain resonance (non-monotonicity) at a frequency that decreases with increasing recovery time constant of synaptic depression (tau(d)). This resonance is mitigated by the presence of F. For D, F and FD, coherence improves as the synaptic conductance time constant (tau(g)) increases, yet the mutual information per spike decreases. The information per spike for D and F follows opposite trends as their respective time constants increase. The broadband but non-monotonic gain and coherence functions seen in vivo suggest that D and perhaps FD dynamics are involved at this synapse. Our results further predict that the likely synaptic configuration is a slower tau(g), e.g. via a mixture of AMPA and NMDA synapses, and a relatively smaller synaptic facilitation time constant (tau(f)) and larger tau(d) (with tau(f) smaller than tau(d) and tau(g)). These results are compatible with known physiology.     

White-noise analysis of nonlinear behavior in an insect sensory neuron: Kernel and cascade approaches

A functional expansion was used to model the relationship between a Gaussian white noise stimulus current and the resulting action potential output in the single sensory neuron of the cockroach femoral tactile spine. A new precise procedure was used to measure the kernels of the functional expansion. Very similar kernel estimates were obtained from separate sections of the data produced by the same neuron with the same input noise power level, although some small time-varying effects were detectable in moving through the data. Similar kernel estimates were measured using different input noise power levels for a given cell, or when comparing different cells under similar stimulus conditions. The kernels were used to identify a model for sensory encoding in the neuron, comprising a cascade of dynamic linear, static nonlinear, and dynamic linear elements. Only a single slice of the estimated experimental second-order kernel was used in identifying the cascade model. However, the complete second-order kernel of the cascade model closely resembled the estimated experimental kernel. Moreover, the model could closely predict the experimental action potential train obtained with novel white noise inputs.     

Modeling synaptic transmission of the tripartite synapse

The tripartite synapse denotes the junction of a pre- and postsynaptic neuron modulated by a synaptic astrocyte. Enhanced transmission probability and frequency of the postsynaptic current-events are among the significant effects of the astrocyte on the synapse as experimentally characterized by several groups. In this paper we provide a mathematical framework for the relevant synaptic interactions between neurons and astrocytes that can account quantitatively for both the astrocytic effects on the synaptic transmission and the spontaneous postsynaptic events. Inferred from experiments, the model assumes that glutamate released by the astrocytes in response to synaptic activity regulates store-operated calcium in the presynaptic terminal. This source of calcium is distinct from voltage-gated calcium influx and accounts for the long timescale of facilitation at the synapse seen in correlation with calcium activity in the astrocytes. Our model predicts the inter-event interval distribution of spontaneous current activity mediated by a synaptic astrocyte and provides an additional insight into a novel mechanism for plasticity in which a low fidelity synapse gets transformed into a high fidelity synapse via astrocytic coupling.     

Parametric and non-parametric modeling of short-term synaptic plasticity. Part II: Experimental study

This paper presents a synergistic parametric and non-parametric modeling study of short-term plasticity (STP) in the Schaffer collateral to hippocampal CA1 pyramidal neuron (SC) synapse. Parametric models in the form of sets of differential and algebraic equations have been proposed on the basis of the current understanding of biological mechanisms active within the system. Non-parametric Poisson–Volterra models are obtained herein from broadband experimental input–output data. The non-parametric model is shown to provide better prediction of the experimental output than a parametric model with a single set of facilitation/depression (FD) process. The parametric model is then validated in terms of its input–output transformational properties using the non-parametric model since the latter constitutes a canonical and more complete representation of the synaptic nonlinear dynamics. Furthermore, discrepancies between the experimentally-derived non-parametric model and the equivalent non-parametric model of the parametric model suggest the presence of multiple FD processes in the SC synapses. Inclusion of an additional set of FD process in the parametric model makes it replicate better the characteristics of the experimentally-derived non-parametric model. This improved parametric model in turn provides the requisite biological interpretability that the non-parametric model lacks.     

Role of temperature in quanta mechanisms of facilitation in the frog neuromuscular junction

The results of computer simulations on the Double Barrier Synapse (DBS) model are presented which quantify the relationship between the synapse parameters and the quanta transfer process. The DBS model is applicable to a variety of states of synaptic activity, and by changing the synapse parameters it is possible to simulate various conditions of quanta transmission. The influence of the bathing solution temperature change on the synaptic parameters under different conditions of transmitter release in the frog neuromuscular junction is investigated. Simulations demonstrate that several synaptic parameters, including the parameters of the presynaptic membrane, are not affected by the temperature change. It is shown that a stimulation frequency exists at which the steady-state level of facilitation during a long train of stimuli is the same for a wide range of temperatures. Received: 2 August 1996 / Accepted in revised form: 19 February 1998     

Single neuron dynamics and computation

At the single neuron level, information processing involves the transformation of input spike trains into an appropriate output spike train. Building upon the classical view of a neuron as a threshold device, models have been developed in recent years that take into account the diverse electrophysiological make-up of neurons and accurately describe their input-output relations. Here, we review these recent advances and survey the computational roles that they have uncovered for various electrophysiological properties, for dendritic arbor anatomy as well as for short-term synaptic plasticity.     

Nonlinear signal transmission between second- and third-order neurons of cockroach ocelli

Transfer characteristics of the synapse made from second- to third-order neurons of cockroach ocelli were studied using simultaneous microelectrode penetrations and the application of tetrodotoxin. Potential changes were evoked in second-order neurons by either an extrinsic current or a sinusoidally modulated light. The synapse had a low-pass filter characteristic with a cutoff frequency of 25-30 Hz, which passed most presynaptic signals. The synapse operated at an exponentially rising part of the overall sigmoidal input/output curve relating pre- and postsynaptic voltages. Although the response of the second-order neuron to sinusoidal light was essentially linear, the response of the third-order neuron contained an accelerating nonlinearity: the response amplitude was a positively accelerated function of the stimulus contrast, reflecting nonlinear synaptic transmission. The response of the third-order neuron exhibited a half-wave rectification: the depolarizing response to light decrement was much larger than the hyperpolarizing response to light increment. Nonlinear synaptic transmission also enhanced the transient response to step-like intensity changes. I conclude that (a) the major function of synaptic transmission between second- and third-order neurons of cockroach ocelli is to convert linear presynaptic signals into nonlinear ones and that (b) signal transmission at the synapse between second- and third-order neurons of cockroach ocelli fundamentally differs from that at the synapse between photoreceptors and second-order neurons of visual systems so far studied, where the synapse operates in the midregion of the characteristic curve and the transmission is essentially linear.     

Effects of active conductance distribution over dendrites on the synaptic integration in an identified nonspiking interneuron

The synaptic integration in individual central neuron is critically affected by how active conductances are distributed over dendrites. It has been well known that the dendrites of central neurons are richly endowed with voltage- and ligand-regulated ion conductances. Nonspiking interneurons (NSIs), almost exclusively characteristic to arthropod central nervous systems, do not generate action potentials and hence lack voltage-regulated sodium channels, yet having a variety of voltage-regulated potassium conductances on their dendritic membrane including the one similar to the delayed-rectifier type potassium conductance. It remains unknown, however, how the active conductances are distributed over dendrites and how the synaptic integration is affected by those conductances in NSIs and other invertebrate neurons where the cell body is not included in the signal pathway from input synapses to output sites. In the present study, we quantitatively investigated the functional significance of active conductance distribution pattern in the spatio-temporal spread of synaptic potentials over dendrites of an identified NSI in the crayfish central nervous system by computer simulation. We systematically changed the distribution pattern of active conductances in the neuron's multicompartment model and examined how the synaptic potential waveform was affected by each distribution pattern. It was revealed that specific patterns of nonuniform distribution of potassium conductances were consistent, while other patterns were not, with the waveform of compound synaptic potentials recorded physiologically in the major input-output pathway of the cell, suggesting that the possibility of nonuniform distribution of potassium conductances over the dendrite cannot be excluded as well as the possibility of uniform distribution. Local synaptic circuits involving input and output synapses on the same branch or on the same side were found to be potentially affected under the condition of nonuniform distribution while operation of the major input-output pathway from the soma side to the one on the opposite side remained the same under both conditions of uniform and nonuniform distribution of potassium conductances over the NSI dendrite.     

Is the movement representation in the motor cortex a moving target?

In redundant neural networks, many different combinations of connection weights will produce the same output, thereby providing many possible solutions for a given computation. In this issue of Neuron, Rokni et al. propose that the arm movement representations in the cerebral cortex act like redundant networks that drift randomly between different synaptic configurations with equivalent input-output behavior because of random noise in the adaptive learning mechanism.     

Targeting Synaptic Dysfunction in Alzheimer’s Disease Therapy

In the past years, major efforts have been made to understand the genetics and molecular pathogenesis of Alzheimer??s disease (AD), which has been translated into extensive experimental approaches aimed at slowing down or halting disease progression. Advances in transgenic (Tg) technologies allowed the engineering of different mouse models of AD recapitulating a range of AD-like features. These Tg models provided excellent opportunities to analyze the bases for the temporal evolution of the disease. Several lines of evidence point to synaptic dysfunction as a cause of AD and that synapse loss is a pathological correlate associated with cognitive decline. Therefore, the phenotypic characterization of these animals has included electrophysiological studies to analyze hippocampal synaptic transmission and long-term potentiation, a widely recognized cellular model for learning and memory. Transgenic mice, along with non-Tg models derived mainly from exogenous application of A??, have also been useful experimental tools to test the various therapeutic approaches. As a result, numerous pharmacological interventions have been reported to attenuate synaptic dysfunction and improve behavior in the different AD models. To date, however, very few of these findings have resulted in target validation or successful translation into disease-modifying compounds in humans. Here, we will briefly review the synaptic alterations across the different animal models and we will recapitulate the pharmacological strategies aimed at rescuing hippocampal plasticity phenotypes. Finally, we will highlight intrinsic limitations in the use of experimental systems and related challenges in translating preclinical studies into human clinical trials.